グラフ理論に基づいた正則な係数行列の構築によるマトリックス法LCIの一般化
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Background, Aim, and Scope. Matrix-based life cycle inventory (LCI) analysis can be effectively used to evaluate the environmental impact of a product system including closed loops (e.g. reuse and recycling). However, the matrix-based method has not yet been widely used because most life cycle assessment (LCA) practitioners find it complicated and difficult to satisfactorily construct a regular coefficient matrix from a large amount of input/output data collected from all the processes that compose a product system. The authors aim to develop a method to construct a regular coefficient matrix, which would enable the widespread use of the matrix method. The method proposed in the authors' previous paper can be used to redescribe a product system as a geometrical figure and construct a regular coefficient matrix using graph theory. However, the method cannot be used when the geometrical figure corresponding to a product system is a non-planar graph. The objective of the present study is to improve and generalize the method so that it can be used with an arbitrary product system.<BR>Methods. First, a product system is redescribed as a geometrical figure by using the five basic components as is done in the previous paper. The redescribed geometrical figure can be expressed as a graph that consists of abstract concepts of nodes and edges in the graph theory. Then, a coefficient matrix for the product system is constructed from the constraint conditions of the product system (e.g. the balance of energy and materials). Because the coefficient matrix must be a regular matrix, the constraint conditions are required to be linearly independent of each other. A combination of linearly independent equations can be found by using a spanning tree that is constructed from the graph corresponding to the product system. A spanning tree can be constructed from every connected graph irrespective of whether the connected graph is planar or not; therefore, a regular coefficient matrix can be constructed from an arbitrary product system. <BR>Results and Discussion. A general method was developed to construct a regular coefficient matrix for an arbitrary product system based on graph theory. The validity of the developed method was demonstrated by using two simple numerical examples. The authors would like to emphasize that the developed method contributes to the generalization and the widespread use of the matrix-based LCI analysis. The generalized algorithm proposed in the present study enables us to construct a regular coefficient matrix and carry out LCI analysis using computer software. The authors plan to create such software, which would enable LCA practitioners to easily carry out the matrix-based LCI analysis without having to manually construct a coefficient matrix.
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